CN103926881A - Speed-fluctuation-free parameter curve direct interpolation method based on secant method - Google Patents

Abstract

Provided is a speed-fluctuation-free parameter curve direct interpolation method based on a secant method. The method comprises the seven steps that in the direct interpolation of a parameter curve, the secant method replaces a traditional string method, and a continuous secant section is used for approaching the parameter curve to be interpolated. The speed-fluctuation-free parameter curve direct interpolation method based on the secant method is characterized in that the interpolation secant section is formed in the interpolation process, namely the interpolation straight line section formed at each interpolation period and the parameter curve to be interpolated are intersected to form a secant. The speed-fluctuation-free parameter curve direct interpolation method based on the secant method can achieve the speed-fluctuation-free interpolation, and can achieve higher interpolation precision than other parameter curve direct interpolation methods under the restraint of the same geometric error, and meanwhile due to the fact that the parameter curve direct interpolation method eliminates the speed fluctuation, the smooth feed speed planning is simpler, and the efficiency of an interpolator is improved. The method is applicable to a parameter curve interpolator of a high-speed and high-precision numerical control system.

Description

A kind of based on secant method without velocity perturbation parametric line direct interpolation method

Technical field

The present invention relates to a kind of based on secant method without velocity perturbation parametric line direct interpolation method, belong to the digital control processing technique field of digital control system.

Background technology

Existing existing parametric line direct interpolation method can not be eliminated the impact of Interpolation Process medium velocity fluctuation completely.Velocity perturbation, by instruction speed of feed and the inconsistent generation of actual feed, causes actual feed to fluctuate up and down in instruction speed of feed, can weigh by velocity perturbation rate, as shown in the formula:

$\mathrm{\σ}=\left|\frac{V_a-V_c}{V_c}\right|\×100\%$

Wherein σ is velocity perturbation rate, V _afor actual feed, V _cfor instruction speed of feed.Velocity perturbation meeting makes the level and smooth speed of feed planning of interpolation become difficult, causes actual interpolation path and path planning inconsistent, therefore can affect the interpolation precision of interpolator, even causes flutter.The impact how release rate fluctuation brings is the key point in parametric line direct interpolation always.

Parametric line direct interpolation method is the earliest even parameter interpolation method, and the parameter increase of each interpolation cycle is normal value.This method is calculated simple, but constant parameter increase cannot be determined speed of feed, therefore easily to lathe, impacts.Occurred afterwards Constant feeding rate interpolation, and occurred that feedback interpolating method carrys out the size of control rate fluctuation, but this method is not considered the impact of geometric error factor, so machining precision cannot guarantee.Along with the development of technology, velocity perturbation, geometric error, dynamics parameter, feed system dynamic perfromance etc. are progressively used as in the direct interpolation process that constraint takes into account parametric line, and parametric line direct interpolation method is day by day perfect.By the computing method of interpolation parameters, parametric line direct interpolation method can be divided three classes: direct numerical method, feedback numerical method and arc length-parameter fitting method.Directly numerical method is the basis of parametric line direct interpolation method, as Taylor single order, the second order method of development, and Runge-Kutta method etc., the velocity perturbation rate that this method produces is uncontrollable.Feedback direct computing method has increased the constraint of velocity perturbation rate on the basis of direct numerical method, by the method for iteration, calculate interpolation parameters increment until velocity perturbation rate meets the demands, the size of energy ACTIVE CONTROL velocity perturbation rate, but release rate fluctuation completely, and the increase of iterations has reduced the real-time performance of interpolator.Arc length-parameter fitting method is the piecewise polynomial function of structure arc length-parameter, though size that cannot ACTIVE CONTROL velocity perturbation rate, the stability bandwidth that can effectively underspeed, yet the impact that still release rate fluctuates and brings completely.Such as the interpolating method based on arc length compensation and feedback, speed adaptive interpolating method etc. occurring afterwards, can effectively reduce iterations and improve interpolation efficiency, but all release rate fluctuations completely.

Different from traditional parametric line string of a musical instrument method interpolating method, of the present invention a kind of based on secant method without velocity perturbation parametric line direct interpolation method, use secant section to go to approach as interpolation straight-line segment to be interpolated curve, can make the velocity perturbation producing in Interpolation Process is 0.In Interpolation Process, by secant section, as interpolation straight-line segment, can produce radial error and bow high level error, if use, in traditional strings collimation method direct interpolation method, utilize the method for longbow high level error restriction interpolation speed of feed, the maximum geometric error that can use radial error that secant method direct interpolation method produces in Interpolation Process and bow high level error to be far smaller than setting, so interpolation precision can improve greatly with respect to traditional string of a musical instrument method.

In parametric line speed of feed planning process, due to the rounding of interpolation cycle number and the generation of velocity perturbation, tend to cause actual interpolation path and theoretical path planning inconsistent.The rounding of periodicity (fractions omitted part when computation period is counted retains integral part) always makes actual interpolation path be less than theoretical path planning; Velocity perturbation both may make actual interpolation path bigger than normal, may be also that actual interpolation path is less than normal.The periodicity that the speed of feed planning error that the rounding of periodicity causes can simply increase constant-speed operation compensates interpolation path difference, and the impact that velocity perturbation causes often needs certain cross-talk segment of curve to re-start speed planning or use other compensation methodes, thereby improved the difficulty of speed of feed planning.Due to this patent invention, based on secant method, without velocity perturbation parametric line direct interpolation method, eliminated velocity perturbation completely, therefore only need the speed of feed planning error that the rounding of interpolation cycle number is caused simply to compensate, than traditional string of a musical instrument method, greatly reduce the difficulty of speed of feed planning, improved the efficiency of interpolator.

Summary of the invention

The present invention relates to a kind of based on secant method without velocity perturbation parametric line direct interpolation method, its objective is in the situation that not improving interpolation algorithm complexity, can make the velocity perturbation rate in Interpolation Process is 0, the impact that completely release rate fluctuation brings.Meanwhile, under same geometric error constraint, can reach higher interpolation precision, and make interpolation speed of feed planning process become easier.Be applicable to the parametric line direct interpolation device of high-speed, high precision digital control system.

The technical solution used in the present invention is as follows: a kind of based on secant method without velocity perturbation parametric line direct interpolation method, adopt secant section to substitute string of a musical instrument section in existing interpolation algorithm and go to approach and be interpolated curve.First use direct numerical method calculating parameter initial value, then by a kind of Newton iteration method based on redundancy coefficient, ask for the secant section meeting the demands, according to instruction speed of feed and interpolation cycle intercepting secant section, be finally that interpolation secant section fluctuates with complete release rate, Figure 1 shows that method flow diagram of the present invention, Figure 2 shows that an interpolating method schematic diagram in interpolation cycle.

The present invention relates to a kind of based on secant method without velocity perturbation parametric line direct interpolation method, its concrete operation step is as follows:

Step 1: parametric line is C (u)={ x (u), y (u), z (u) }, and u is parameter of curve, general 0≤u≤1.When interpolation starts, establishing the 0th interpolated point parameter is u ₀the=0,0th interpolated point is P ₀=C (u ₀)={ x ₀, y ₀, z ₀.

Step 2: the instruction speed of feed of i interpolation cycle is V _i, interpolation cycle is T _c, computations Interpolation step-length is L _i=V _it _c.

Step 3: use the Taylor second order method of development to calculate the initial value of next interpolation parameters value as follows:

$\left\{\begin{array}{c}{u}_{i+1}^{\left(0\right)}=u_i+\frac{\mathrm{du}}{\mathrm{dt}}{|}_{u_i}{T}_c+\frac{d^{2}u}{{\mathrm{dt}}^2}{|}_{u_i}\frac{{T_c}^2}{2}\\ \frac{\mathrm{du}}{\mathrm{dt}}{|}_{u_i}=V_i/\left|\right|C^{\′}\left(u\right){|}_{u_i}\left|\right|\\ \frac{d^{2}u}{{\mathrm{dt}}^2}{|}_{u_i}=-{V_i}^2\⟨C^{\′}\left(u\right){|}_{u_i},C^{\′\′}\left(u\right){|}_{u_i}\⟩/{\left|\right|C^{\′}\left(u\right){|}_{u_i}\left|\right|}^4\end{array}\right.$

Step 4: current interpolated point P _i={ x _i, y _i, z _ito next interpolation parameters value initial value point on corresponding curve distance be use the next interpolation parameters of a kind of Newton iteration method correction based on redundancy coefficient as follows:

$\left\{\begin{array}{c}\begin{array}{}\end{array}\mathrm{do}:\left\{\begin{array}{c}{\mathrm{\&Delta;L}}_{i+1}^{\left(j\right)}=\left|\right|C\left(u_{i+1}^{\left(j\right)}\right)-P_i\left|\right|\\ F\left(u_{i+1}^{\left(j\right)}\right)=\left|\right|C\left(u_{i+1}^{\left(j\right)}\right)-P_i\left|\right|-(1-\mathrm{\&epsiv;}){\mathrm{\&Delta;L}}_{i+1}^{\left(j\right)}\\ F^{\&prime;}\left(u_{i+1}^{\left(j\right)}\right)=\frac{\&lang;C\left(u_{i+1}^{\left(j\right)}\right)-P_i,C^{\&prime;}\left(u_{i+1}^{\left(j\right)}\right)\&rang;}{\left|\right|C\left(u_{i+1}^{\left(j\right)}\right)-P_i\left|\right|}\\ \end{array}\right.,j=\mathrm{0,1,2},...\\ \mathrm{until}:{\mathrm{\&tau;L}}_i<{\mathrm{\&Delta;L}}_{i+1}^{\left(j\right)}<L_i,\mathrm{then}:u_{i+1}=u_{i+1}^{\left(j\right)}\end{array}\right.$

Wherein ε, τ are redundancy coefficient.ε is controlling convergence of algorithm speed, and τ is controlling secant section with respect to the position that is interpolated curve, and two coefficients need mutually to coordinate guarantee iterative algorithms restrains rapidly, generally gets ε=0.01, τ=0.9 can make algorithm finish in 2 iteration.After iterative algorithm finishes, the final value of next interpolation parameters is u _i+1, the point on corresponding curve is C (u _i+1).

Step 5: current interpolated point P _i={ x _i, y _i, z _ito putting C (u _i+1) unit direction vector be:

${\stackrel{\&RightArrow;}{d}}_i(C\left(u_{i+1}\right)-P_i)/\left|\right|C\left(u_{i+1}\right)-P_i\left|\right|$

This direction vector determines the moving direction of cutter in current interpolation cycle.

Step 6: the next one without velocity perturbation interpolated point is:

$P_{i+1}=\{x_{i+1},y_{i+1},z_{i+1}\}=P_i+L_i{\stackrel{\&RightArrow;}{d}}_i$

Step 7: if u _i+1≤ 1 increases interpolation cycle number is i=i+1, returns to step 2; Otherwise parametric line direct interpolation completes, the P of generation _i={ x _i, y _i, z _i(i=0,1,2 ...) and for Interpolation Process generate without velocity perturbation error interpolated point.

Interpolation Process geometric error is calculated: the interpolation parameters u of i interpolation cycle _ipoint on the curve at place is C (u _i), corresponding is P without velocity perturbation interpolated point _i, radius-of-curvature is ρ _i, and main method arrow is the next one that interpolation obtains through secant method is P without velocity perturbation interpolated point _i+1, according to Fig. 2, can derive and be respectively in the radial error of i interpolation cycle and bow high level error:

$\left\{\begin{array}{c}{e}_i^r=|{\mathrm{\&rho;}}_i-||C\left(u_i\right)+{\mathrm{\&rho;}}_i{\stackrel{\&RightArrow;}{n}}_i-P_i|\left|\right|\\ e_i^c=|{\mathrm{\&rho;}}_i-||C\left(u_i\right)+{\mathrm{\&rho;}}_i{\stackrel{\&RightArrow;}{n}}_i-\frac{P_i+P_{i+1}}{2}|\left|\right|\end{array}\right.$

Wherein, the symbol in each step ' || || ' be straight length calculating, ' <, > is that vectorial scalar product calculates to symbol.

Of the present invention a kind of based on secant method without velocity perturbation parametric line direct interpolation method, its advantage and effect are: the interpolated point that Interpolation Process generates can not produce velocity perturbation, eliminated the adverse effect that velocity perturbation brings completely, as reduced machining precision, cause flutter, increasing speed of feed planning difficulty etc.The present invention can make the speed planning of Interpolation Process become simpler, improves interpolation geometric accuracy and computing velocity, is suitable for high-speed, high precision digital control system.

Accompanying drawing explanation

Fig. 1 is method flow diagram of the present invention;

Fig. 2 is the method schematic diagram of the present invention in an interpolation cycle;

Fig. 3 is exemplifying embodiment 1: the horizontal 8 font nurbs curves of secondary;

Fig. 4 is the horizontal 8 font nurbs curve interpolation feed speed curve of secondary;

Fig. 5 is the horizontal 8 font nurbs curve interpolation iterations curves of secondary;

Fig. 6 is the horizontal 8 font nurbs curve interpolation radial error curves of secondary;

Fig. 7 is the horizontal 8 font nurbs curve interpolation bow high level error curves of secondary;

Fig. 8 is 2: three buttferfly-type nurbs curves of exemplifying embodiment;

Fig. 9 is three buttferfly-type nurbs curve interpolation feed speed curve;

Figure 10 is three buttferfly-type nurbs curve interpolation iterations curves;

Figure 11 is three buttferfly-type nurbs curve interpolation radial error curves;

Figure 12 is three buttferfly-type nurbs curve interpolations bow high level error curve;

In figure, symbol, code name are described as follows:

I (i=0,1,2 ...) be interpolation cycle number;

J (i=0,1,2) is the Newton iterative method number of times based on redundancy coefficient;

U _i(i=0,1,2 ...) be the interpolation parameters of i interpolation cycle;

C(u _i) (i=0,1,2 ...) be the point on the curve that the interpolation parameters of i interpolation cycle is corresponding;

P _i(i=0,1,2 ...) and for the inventive method interpolation generate without velocity perturbation interpolated point;

(i=0,1,2 ...) be interpolation moving direction vector;

(i=0,1,2 ...) be the main method arrow of curve at the interpolation parameters place of i interpolation cycle;

Embodiment

Nurbs curve is the Typical Representative of parametric line, and the nurbs curve of take carries out concrete operations to the present invention as example.The number of times that makes nurbs curve is p, and control fixed-point number is n, and knot vector is U={u _j(0≤j≤m), control vertex is A={a _j(a _j={ x _j, y _j, z _j, 0≤j≤n), weighted vector is W={w _j(0≤j≤n), this nurbs curve can be expressed as:

$C\left(u\right)=\{x\left(u\right),y\left(u\right),z\left(u\right)\}=\frac{\underset{j=0}{\overset{n}{\mathrm{\&Sigma;}}}{N}_{j,p}\left(u\right)a_j{w}_j}{\underset{j=0}{\overset{n}{\mathrm{\&Sigma;}}}{N}_{j,p}\left(u\right)w_j}---\left(1\right)$

Wherein u (0≤u≤1) is parameter of curve, N _j,p(u) be basis function, use following recurrence method to calculate:

The main method of curve is vowed as follows and is calculated:

$\stackrel{\&RightArrow;}{n\left(u\right)}=\frac{C^{\&prime;}\left(u\right)}{\left|\right|C^{\&prime;}\left(u\right)\left|\right|}\&times;\frac{C^{\&prime;}\left(u\right)\&times;C^{\&prime;\&prime;}\left(u\right)}{\left|\right|C^{\&prime;}\left(u\right)\&times;C^{\&prime;\&prime;}\left(u\right)\left|\right|}---\left(2\right)$

The radius-of-curvature of curve is calculated by following:

$\mathrm{\&rho;}\left(u\right)=\frac{{\left|\right|C^{\&prime;}\left(u\right)\left|\right|}^3}{\left|\right|C^{\&prime;}\left(u\right)\&times;C^{\&prime;\&prime;}\left(u\right)\left|\right|}---\left(3\right)$

The present invention relates to a kind of based on secant method without velocity perturbation parametric line direct interpolation method, existing take nurbs curve as parametric line object, operational flowchart is as Fig. 1, the operation chart in each interpolation cycle is as Fig. 2, concrete operation step is as follows:

Step 1: the 0th interpolated point parameter is u ₀=0, it is P that substitution formula (1) obtains the 0th interpolated point ₀=C (u ₀)={ x ₀, y ₀, z ₀.

Step 2: the instruction speed of feed of i interpolation cycle is V _i, interpolation cycle is T _c, computations Interpolation step-length is L _i=V _it _c.

Step 3: use the Taylor second order method of development to calculate the initial value of next interpolation parameters value as follows:

$\left\{\begin{array}{c}{u}_{i+1}^{\left(0\right)}=u_i+\frac{\mathrm{du}}{\mathrm{dt}}{|}_{u_i}{T}_c+\frac{d^{2}u}{{\mathrm{dt}}^2}{|}_{u_i}\frac{{T_c}^2}{2}\\ \frac{\mathrm{du}}{\mathrm{dt}}{|}_{u_i}=V_i/\left|\right|C^{\&prime;}\left(u\right){|}_{u_i}\left|\right|\\ \frac{d^{2}u}{{\mathrm{dt}}^2}{|}_{u_i}=-{V_i}^2\&lang;C^{\&prime;}\left(u\right){|}_{u_i},C^{\&prime;\&prime;}\left(u\right){|}_{u_i}\&rang;/{\left|\right|C^{\&prime;}\left(u\right){|}_{u_i}\left|\right|}^4\end{array}\right.$

Step 4: will bringing formula (1) into obtains calculate itself and current interpolated point P _idistance be the next interpolation parameters of the Newton iteration method correction of use based on redundancy coefficient is as follows:

$\left\{\begin{array}{c}\begin{array}{}\end{array}\mathrm{do}:\left\{\begin{array}{c}{\mathrm{\&Delta;L}}_{i+1}^{\left(j\right)}=\left|\right|C\left(u_{i+1}^{\left(j\right)}\right)-P_i\left|\right|\\ F\left(u_{i+1}^{\left(j\right)}\right)=\left|\right|C\left(u_{i+1}^{\left(j\right)}\right)-P_i\left|\right|-(1-\mathrm{\&epsiv;}){\mathrm{\&Delta;L}}_{i+1}^{\left(j\right)}\\ F^{\&prime;}\left(u_{i+1}^{\left(j\right)}\right)=\frac{\&lang;C\left(u_{i+1}^{\left(j\right)}\right)-P_i,C^{\&prime;}\left(u_{i+1}^{\left(j\right)}\right)\&rang;}{\left|\right|C\left(u_{i+1}^{\left(j\right)}\right)-P_i\left|\right|}\\ \end{array}\right.,j=\mathrm{0,1,2},...\\ \mathrm{until}:{\mathrm{\&tau;L}}_i<{\mathrm{\&Delta;L}}_{i+1}^{\left(j\right)}<L_i,\mathrm{then}:u_{i+1}=u_{i+1}^{\left(j\right)}\end{array}\right.$

Wherein ε, τ are redundancy coefficient, get ε=0.01, τ=0.9.After iterative algorithm finishes, the final value of next interpolation parameters is u _i+1, substitution formula (1) obtains C (u _i+1).

Step 5: current interpolated point P _i={ x _i, y _i, z _ito putting C (u _i+1) unit direction vector be:

${\stackrel{\&RightArrow;}{d}}_i(C\left(u_{i+1}\right)-P_i)/\left|\right|C\left(u_{i+1}\right)-P_i\left|\right|$

This direction vector determines the moving direction of cutter in current interpolation cycle.

Step 6: the next one without velocity perturbation interpolated point is:

$P_{i+1}=\{x_{i+1},y_{i+1},z_{i+1}\}=P_i+L_i{\stackrel{\&RightArrow;}{d}}_i$

Step 7: if u _i+1≤ 1 increases interpolation cycle number is i=i+1, returns to step 2; Otherwise parametric line direct interpolation completes, the P of generation _i={ x _i, y _i, z _i(i=0,1,2, ") be that Interpolation Process generates without velocity perturbation error interpolated point.

Interpolation Process geometric error is calculated: the interpolation parameters of i interpolation cycle is u _i, the main method of use formula (2) calculated curve is vowed and is the radius-of-curvature of use formula (3) calculated curve is ρ _i, radial error and the bow high level error at i interpolation cycle is respectively:

$\left\{\begin{array}{c}{e}_i^r=|{\mathrm{\&rho;}}_i-||C\left(u_i\right)+{\mathrm{\&rho;}}_i{\stackrel{\&RightArrow;}{n}}_i-P_i|\left|\right|\\ e_i^c=|{\mathrm{\&rho;}}_i-||C\left(u_i\right)+{\mathrm{\&rho;}}_i{\stackrel{\&RightArrow;}{n}}_i-\frac{P_i+P_{i+1}}{2}|\left|\right|\end{array}\right.$

Wherein, the symbol in above-mentioned formula ' || || ' for straight length calculates, and symbol ' <, > ' is that vectorial scalar product calculates, symbol ' * ' is that vectorial vector product is calculated.

In order to verify more intuitively, provide actual effect of the present invention two concrete nurbs curve examples below and carry out actual effect checking.

Exemplifying embodiment 1:

Fig. 3 is the horizontal 8 font nurbs curves of a secondary, and its parameter of curve is as follows:

Nodal value: 0,0,0,0.25,0.5,0.5,0.75,1,1,1;

Weights: 1,25,25,1,25,25,1;

Reference mark: (0,0), (120 ,-120), (120,120), (0,0), (120 ,-120), (120,120), (0,0);

Bring parameter of curve into direct interpolation that each implementation step recited above completes nurbs curve, interpolation result is as shown in Fig. 4～7.Fig. 4 is level and smooth interpolation feed speed curve, the constraint by geometric error and dynamics make interpolation speed of feed at some high curvature areas low speed by guarantee interpolation precision and dynamics requirement.Fig. 5 is the iterations that the Newton iteration method based on redundancy surplus is asked rational interpolation secant section, when redundancy surplus coefficient ε=0.01, τ=0.9 o'clock, iterations is 2 times to the maximum, and (0 representative does not need to carry out iteration, interpolation parameters iterative initial value meets the requirements), so iterative algorithm can be obtained interpolation secant section fast.Fig. 6 and Fig. 7 are radial error and the bow high level errors producing in Interpolation Process, when geometric error is constrained to 1 μ m, in figure, can see, the maximum radial error that secant method interpolating method produces and longbow high level error are 0.035 μ m and 0.11 μ m, the maximal value allowing much smaller than geometric error, interpolation precision is high.Due to real-time interpolation process be by this patent invention based on secant method without velocity perturbation parametric line direct interpolation method, carry out interpolation, so the velocity perturbation rate producing in Interpolation Process is 0, eliminated the adverse effect that velocity perturbation brings completely.

Exemplifying embodiment 2:

Fig. 8 is three buttferfly-type nurbs curves, and its parameter of curve is as follows:

Nodal value: 0, 0, 0, 0, 0.0083, 0.0150, 0.0361, 0.0855, 0.1293, 0.1509, 0.1931, 0.2273, 0.2435, 0.2561, 0.2692, 0.2889, 0.3170, 0.3316, 0.3482, 0.3553, 0.3649, 0.3837, 0.4005, 0.4269, 0.4510, 0.4660, 0.4891, 0.5000, 0.5109, 0.5340, 0.5489, 0.5731, 0.5994, 0.6163, 0.6351, 0.6447, 0.6518, 0.6683, 0.6830, 0.7111, 0.7307, 0.7439, 0.7565, 0.7729, 0.8069, 0.8491, 0.8707, 0.9145, 0.9639, 0.9850, 0.9917, 1, 1, 1, 1,

Weights: 1,1,1,1.2,1,1,1,1,1,1,1,2,1,1,5,3,1,1.1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1.1,1,3,5,1,1,2,1,1,1,1,1,1,1,1.2,1,1,1;

Reference mark: (54.493, 52.139), (55.507, 52.139), (56.082, 49.615), (56.780, 44.971), (69.575, 51.358), (77.786, 58.573), (90.526, 67.081), (105.973, 63.801), (100.400, 47.326), (94.567, 39.913), (92.369, 30.485), (83.440, 33.757), (91.892, 28.509), (89.444, 20.393), (83.218, 15.446), (87.621, 4.830), (80.945, 9.267), (79.834, 14.535), (76.074, 8.522), (70.183, 12.550), (64.171, 16.865), (59.993, 22.122), (55.680, 36.359), (56.925, 24.995), (59.765, 19.828), (54.493, 14.940), (49.220, 19.828), (52.060, 24.994), (53.305, 36.359), (48.992, 22.122), (44.814, 16.865), (38.802, 12.551), (32.911, 8.521), (29.152, 14.535), (28.040, 9.267), (21.364, 4.830), (25.768, 15.447), (19.539, 20.391), (17.097, 28.512), (25.537, 33.750), (16.602, 30.496), (14.199, 39.803), (8.668, 47.408), (3.000, 63.794), (18.465, 67.084), (31.197, 58.572), (39.411, 51.358), (52.204, 44.971), (52.904, 49.614), (53.478, 52.139), (54.492, 52.139),

Bring parameter of curve into direct interpolation that each implementation step recited above completes nurbs curve, interpolation result is as shown in Fig. 9～12.Fig. 9 is level and smooth interpolation feed speed curve, the constraint by geometric error and dynamics make interpolation speed of feed at some high curvature areas low speed by guarantee interpolation precision and dynamics requirement.Figure 10 is the iterations that the Newton iteration method based on redundancy surplus is asked rational interpolation secant section, when redundancy surplus coefficient ε=0.01, τ=0.9 o'clock, iterations is 2 times to the maximum, and (0 representative does not need to carry out iteration, interpolation parameters iterative initial value meets the requirements), so iterative algorithm can be obtained interpolation secant section fast.Figure 11 and Figure 12 are radial error and the bow high level errors producing in Interpolation Process, when geometric error is constrained to 1 μ m, in figure, can see, the maximum radial error that secant method interpolating method produces and longbow high level error are 0.1 μ m and 0.5 μ m, the maximal value allowing much smaller than geometric error, interpolation precision is high.Due to real-time interpolation process be by this patent invention based on secant method without velocity perturbation parametric line direct interpolation method, carry out interpolation, so the velocity perturbation rate producing in Interpolation Process is 0, eliminated the adverse effect that velocity perturbation brings completely.

Claims (1)

1. Based on secant method without a velocity perturbation parametric line direct interpolation method, it is characterized in that: the method concrete steps are as follows:

   Step 1: parametric line is C (u)={ x (u), y (u), z (u) }, and u is parameter of curve, 0≤u≤1; When interpolation starts, establishing the 0th interpolated point parameter is u ₀the=0,0th interpolated point is P ₀=C (u ₀)={ x ₀, y ₀, z ₀;

   Step 2: the instruction speed of feed of i interpolation cycle is V _i, interpolation cycle is T _c, computations Interpolation step-length is L _i=V _it _c;

   Step 3: use the Taylor second order method of development to calculate the initial value of next interpolation parameters value as follows:

   $\left\{\begin{array}{c}{u}_{i+1}^{\left(0\right)}=u_i+\frac{\mathrm{du}}{\mathrm{dt}}{|}_{u_i}{T}_c+\frac{d^{2}u}{{\mathrm{dt}}^2}{|}_{u_i}\frac{{T_c}^2}{2}\\ \frac{\mathrm{du}}{\mathrm{dt}}{|}_{u_i}=V_i/\left|\right|C^{\&prime;}\left(u\right){|}_{u_i}\left|\right|\\ \frac{d^{2}u}{{\mathrm{dt}}^2}{|}_{u_i}=-{V_i}^2\&lang;C^{\&prime;}\left(u\right){|}_{u_i},C^{\&prime;\&prime;}\left(u\right){|}_{u_i}\&rang;/{\left|\right|C^{\&prime;}\left(u\right){|}_{u_i}\left|\right|}^4\end{array}\right.$

   Step 4: current interpolated point P _i={ x _i, y _i, z _ito next interpolation parameters value initial value point on corresponding curve distance be use the next interpolation parameters of a kind of Newton iteration method correction based on redundancy coefficient as follows:

   $\left\{\begin{array}{c}\begin{array}{}\end{array}\mathrm{do}:\left\{\begin{array}{c}{\mathrm{\&Delta;L}}_{i+1}^{\left(j\right)}=\left|\right|C\left(u_{i+1}^{\left(j\right)}\right)-P_i\left|\right|\\ F\left(u_{i+1}^{\left(j\right)}\right)=\left|\right|C\left(u_{i+1}^{\left(j\right)}\right)-P_i\left|\right|-(1-\mathrm{\&epsiv;}){\mathrm{\&Delta;L}}_{i+1}^{\left(j\right)}\\ F^{\&prime;}\left(u_{i+1}^{\left(j\right)}\right)=\frac{\&lang;C\left(u_{i+1}^{\left(j\right)}\right)-P_i,C^{\&prime;}\left(u_{i+1}^{\left(j\right)}\right)\&rang;}{\left|\right|C\left(u_{i+1}^{\left(j\right)}\right)-P_i\left|\right|}\\ \end{array}\right.,j=\mathrm{0,1,2},...\\ \mathrm{until}:{\mathrm{\&tau;L}}_i<{\mathrm{\&Delta;L}}_{i+1}^{\left(j\right)}<L_i,\mathrm{then}:u_{i+1}=u_{i+1}^{\left(j\right)}\end{array}\right.$

   Wherein ε, τ are redundancy coefficient, ε is controlling convergence of algorithm speed, τ is controlling secant section with respect to the position that is interpolated curve, and two coefficients need mutually to coordinate guarantee iterative algorithms restrains rapidly, gets ε=0.01, τ=0.9 can make algorithm finish in 2 iteration; After iterative algorithm finishes, the final value of next interpolation parameters is u _i+1, the point on corresponding curve is C (u _i+1);

   Step 5: current interpolated point P _i={ x _i, y _i, z _ito putting C (u _i+1) unit direction vector be:

   ${\stackrel{\&RightArrow;}{d}}_i(C\left(u_{i+1}\right)-P_i)/\left|\right|C\left(u_{i+1}\right)-P_i\left|\right|$

   This direction vector determines the moving direction of cutter in current interpolation cycle;

   Step 6: the next one without velocity perturbation interpolated point is:

   $P_{i+1}=\{x_{i+1},y_{i+1},z_{i+1}\}=P_i+L_i{\stackrel{\&RightArrow;}{d}}_i$

   Step 7: if u _i+1≤ 1 increases interpolation cycle number is i=i+1, returns to step 2; Otherwise parametric line direct interpolation completes, the P of generation _i={ x _i, y _i, z _i(i=0,1,2 ...) and for Interpolation Process generate without velocity perturbation error interpolated point;

   Interpolation Process geometric error is calculated: the interpolation parameters u of i interpolation cycle _ipoint on the curve at place is C (u _i), corresponding is P without velocity perturbation interpolated point _i, radius-of-curvature is ρ _i, and main method arrow is the next one that interpolation obtains through secant method is P without velocity perturbation interpolated point _i+1, derivation is respectively at radial error and the bow high level error of i interpolation cycle:

   $\left\{\begin{array}{c}{e}_i^r=|{\mathrm{\&rho;}}_i-||C\left(u_i\right)+{\mathrm{\&rho;}}_i{\stackrel{\&RightArrow;}{n}}_i-P_i|\left|\right|\\ e_i^c=|{\mathrm{\&rho;}}_i-||C\left(u_i\right)+{\mathrm{\&rho;}}_i{\stackrel{\&RightArrow;}{n}}_i-\frac{P_i+P_{i+1}}{2}|\left|\right|\end{array}\right.$

   Wherein, the symbol in each step ' || || ' be straight length calculating, ' <, > ' is that vectorial scalar product calculates to symbol.